A note on the efficient evaluation of a modified Hilbert transformation

Abstract In this note we consider an efficient data–sparse approximation of a modified Hilbert type transformation as it is used for the space–time finite element discretization of parabolic evolution equations in the anisotropic Sobolev space H1,1/2(Q). The resulting bilinear form of the first-order time derivative is symmetric and positive definite, and similar as the integration by parts formula for the Laplace hypersingular boundary integral operator in 2D. Hence we can apply hierarchical matrices for data–sparse representations and for acceleration of the computations. Numerical results show the efficiency in the approximation of the first-order time derivative. An efficient realisation of the modified Hilbert transformation is a basic ingredient when considering general space–time finite element methods for parabolic evolution equations, and for the stable coupling of finite and boundary element methods in anisotropic Sobolev trace spaces.